Pascal's Triangle and the Binomial Expansion

Name: Pascal's Triangle and the Binomial Expansion
Author: StemPy Lab

Pascal's triangle and the expansion of (a + b) to the power n for a whole number n. The general term, the middle term or terms, and the term that is free of x.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Pascal's triangle and the expansion
The expansion of $(a + b)^n$ has $n + 1$ terms. The powers of $a$ go down from $n$ to $0$ and the powers of $b$ go up from $0$ to $n$ . The coefficients are the numbers in row $n$ of Pascal's triangle, and each is ${}^nC_r$ , the number of ways to choose which $r$ of the $n$ brackets give a $b$ . Replacing $b$ by $-b$ gives $(a - b)^n$ , and putting $a = b = 1$ shows that each row adds up to $2^n$ .
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 25 min
The general term
The term with $r$ copies of $b$ is the general term $T_{r+1} = {}^nC_r\,a^{n-r}b^r$ . It is the $(r + 1)$ th term, because $r = 0$ gives the first. To find a particular term, put in the value of $r$ . To find the term with a given power of $x$ , write that power as an expression in $r$ and set it equal to the power you want. Then solve for $r$ . The answer must be a whole number from $0$ to $n$ , or there is no such term.
1 simulation1 practice question1 Python program3 problems
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Lesson 3about 25 min
Middle terms and the term free of x
The expansion of $(a + b)^n$ has $n + 1$ terms. When $n$ is even there is one middle term, $T_{n/2+1}$ . When $n$ is odd there are two, $T_{(n+1)/2}$ and $T_{(n+3)/2}$ . The term free of $x$ has $x^0$ , so its $r$ makes the power of $x$ zero. For a small $x$ , a few terms of $(1 + x)^n$ give a good estimate, because the powers of $x$ shrink fast.
1 simulation1 practice question1 Python program4 problems1 Python lab1 question set1 note
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Pascal's Triangle and the Binomial Expansion: Practice Paper

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Pascal's Triangle and the Binomial Expansion

Course outline (3 lessons)

Pascal's triangle and the expansion

The general term

Middle terms and the term free of x

Test Your Learning

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